Extreme Palindromes

dominated by the corresponding segment of the graph of f . While the domain of the function g is convex by the induction hypothesis, one is led to wonder about the concavity of g. Again, a more general result sheds light here. Let c be a real number and g1, . . . , gn be concave functions on a vector space V . Suppose that S ⊆ V n is a convex set, and define g : S −→ R by g(v) = c + nj=1 g j (v j ) for all v = (v1, . . . , vn) ∈ S. Then g is a concave function. For the proof, it suffices to show that g − c is concave, since adding a constant to a concave function preserves concavity. So suppose that u = (u1, . . . , un), v = (v1, . . . , vn) ∈ S, and t ∈ [0, 1]. Then